58 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Thin Film Equations with Nonlinear Deterministic and Stochastic Perturbations
In this paper we consider stochastic thin-film equation with nonlinear drift
terms, colored Gaussian Stratonovych noise, as well as nonlinear colored Wiener
noise. By means of Trotter-Kato-type decomposition into deterministic and
stochastic parts, we couple both of these dynamics via a discrete-in-time
scheme, and establish its convergence to a non-negative weak martingale
solution
Depleting the signal: Analysis of chemotaxis-consumption modelsâA survey
We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures
Local existence of solutions and comparison principle for initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory
We consider an initial value problem for a nonlinear parabolic equation with
memory under nonlinear nonlocal boundary condition. In this paper we study
classical solutions. We establish the existence of a local maximal solution. It
is shown that under some conditions a supersolution is not less than a
subsolution. We find conditions for the positiveness of solutions. As a
consequence of the positiveness of solutions and the comparison principle of
solutions, we prove the uniqueness theorem
Mathematical Analysis of Charge and Heat Flow in Organic Semiconductor Devices
Organische Halbleiterbauelemente sind eine vielversprechende Technologie, die das Spektrum der optoelektronischen Halbleiterbauelemente erweitert und etablierte Technologien basierend auf anorganischen Halbleitermaterialien ersetzen kann. FĂŒr Display- und Beleuchtungsanwendungen werden sie z. B. als organische Leuchtdioden oder Transistoren verwendet. Eine entscheidende Eigenschaft organischer Halbleitermaterialien ist, dass die Ladungstransporteigenschaften stark von der Temperatur im Bauelement beeinflusst werden. Insbesondere nimmt die elektrische LeitfĂ€higkeit mit der Temperatur zu, so dass Selbsterhitzungseffekte, einen groĂen Einfluss auf die Leistung der Bauelemente haben. Mit steigender Temperatur nimmt die elektrische LeitfĂ€higkeit zu, was wiederum zu gröĂeren Strömen fĂŒhrt. Dies fĂŒhrt jedoch zu noch höheren Temperaturen aufgrund von Joulescher WĂ€rme oder RekombinationswĂ€rme. Eine positive RĂŒckkopplung liegt vor. Im schlimmsten Fall fĂŒhrt dieses Verhalten zum thermischen Durchgehen und zur Zerstörung des Bauteils. Aber auch ohne thermisches Durchgehen fĂŒhren Selbsterhitzungseffekte zu interessanten nichtlinearen PhĂ€nomenen in organischen Bauelementen, wie z. B. die S-förmige Beziehung zwischen Strom und Spannung. In Regionen mit negativem differentiellen Widerstand fĂŒhrt eine Verringerung der Spannung ĂŒber dem Bauelement zu einem Anstieg des Stroms durch das Bauelement. Diese Arbeit soll einen Beitrag zur mathematischen Modellierung, Analysis und numerischen Simulation von organischen Bauteilen leisten. Insbesondere wird das komplizierte Zusammenspiel zwischen dem Fluss von LadungstrĂ€gern (Elektronen und Löchern) und WĂ€rme diskutiert. Die zugrundeliegenden Modellgleichungen sind Thermistor- und Energie-Drift-Diffusion-Systeme. Die numerische Diskretisierung mit robusten hybriden Finite-Elemente-/Finite-Volumen-Methoden und Pfadverfolgungstechniken zur Erfassung der in Experimenten beobachteten S-förmigen Strom-Spannungs-Charakteristiken wird vorgestellt.Organic semiconductor devices are a promising technology to extend the range of optoelectronic semiconductor devices and to some extent replace established technologies based on inorganic semiconductor materials. For display and lighting applications, they are used as organic light-emitting diodes (OLEDs) or transistors. One crucial property of organic semiconductor materials is that charge-transport properties are heavily influenced by the temperature in the device. In particular, the electrical conductivity increases with temperature, such that self-heating effects caused by the high electric fields and strong recombination have a potent impact on the performance of devices. With increasing temperature, the electrical conductivity rises, which in turn leads to larger currents. This, however, results in even higher temperatures due to Joule or recombination heat, leading to a feedback loop. In the worst case, this loop leads to thermal runaway and the complete destruction of the device. However, even without thermal runaway, self-heating effects give rise to interesting nonlinear phenomena in organic devices, like the S-shaped relation between current and voltage resulting in regions where a decrease in voltage across the device results in an increase in current through it, commonly denoted as regions of negative differential resistance. This thesis aims to contribute to the mathematical modeling, analysis, and numerical simulation of organic semiconductor devices. In particular, the complicated interplay between the flow of charge carriers (electrons and holes) and heat is discussed. The underlying model equations are of thermistor and energy-drift-diffusion type. Moreover, the numerical approximation using robust hybrid finite-element/finite-volume methods and path-following techniques for capturing the S-shaped current-voltage characteristics observed in experiments are discussed
Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
We establish boundedness estimates for solutions of generalized porous medium equations of the form âtu+(âL)[um]=0in RNĂ(0,T), where mâ„1 and âL is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, LĂ©vy operators. Our quantitative estimates take the form of precise L1âLâ-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of âL and IâL. In the linear case m=1, it is well-known that the L1âLâ-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting m>1. First, we can show that operators for which ultracontractivity holds, also provide L1âLâ-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by 0-order LĂ©vy operators like âL=IâJâ. They do not regularize when m=1, but we show that surprisingly enough they do so when m>1, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iterationJ. Endal has received funding from the European Unionâs Horizon 2020 research and
innovation programme under the Marie SkĆodowska-Curie grant agreement no. 839749
âNovel techniques for quantitative behaviour of convection-diffusion equations (techFRONT)â, and from the Research Council of Norway under the MSCA-TOPP-UT grant
agreement no. 312021.
M. Bonforte was partially supported by the Projects MTM2017-85757-P and PID2020-
113596GB-I00 (Spanish Ministry of Science and Innovation). M. Bonforte moreover
acknowledges financial support from the Spanish Ministry of Science and Innovation,
through the âSevero Ochoa Programme for Centres of Excellence in R&Dâ (CEX2019-
000904-S) and by the European Unionâs Horizon 2020 research and innovation programme under the Marie SkĆodowska-Curie grant agreement no. 77782
Analysis of Reaction-Diffusion Models with the Taxis Mechanism
This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics
Dedication to Professor Michael Tribelsky
Professor Tribelsky's accomplishments are highly appreciated by the international community. The best indications of this are the high citation rates of his publications, and the numerous awards and titles he has received. He has made numerous fundamental contributions to an extremely broad area of physics and mathematics, including (but not limited to) quantum solid-state physics, various problems in lightâmatter interaction, liquid crystals, physical hydrodynamics, nonlinear waves, pattern formation in nonequilibrium systems and transition to chaos, bifurcation and probability theory, and even predictions of the dynamics of actual market prices. This book presents several extensions of his results, based on his inspiring publications
Theoretical Concepts of Quantum Mechanics
Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers
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